A072438 -id:A072438 - cp's OEIS frontend

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127

Offset: 1

N. J. A. Sloane

nonn

Also numbers with no prime factors of form 4*k+1.
m is a term iff A072438(m) = m.
Density 0. - Charles R Greathouse IV, Apr 16 2012
Closed under multiplication. Primitive elements are A045326, 2 and the primes of form 4*k+3. - Jean-Christophe Hervé, Nov 17 2013

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Evan M. Bailey, Table of n, a(n) for n = 1..20000 (Terms 1..1000 from T. D. Noe)
Evan M. Bailey, a004144.cpp.
Steven R. Finch, Landau-Ramanujan Constant [Broken link]
Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
Daniel Shanks, Non-hypotenuse numbers, Fib. Quart., Vol. 13, No. 4 (1975), pp. 319-321.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for sequences related to sums of squares.

Complement of A009003.
Cf. A000290, A002145, A005089, A072437, A244659, A244662.
The subsequence of primes is A045326.

import Data.List (elemIndices)
a004144 n = a004144_list !! (n-1)
a004144_list = map (+ 1) $ elemIndices 0 a005089_list
-- Reinhard Zumkeller, Jan 07 2013

fQ[n_] := If[n > 1, First@ Union@ Mod[ First@# & /@ FactorInteger@ n, 4] != 1, True]; Select[ Range@ 127, fQ]
A004144 = Select[Range[127],Length@Reduce[s^2 + t^2 == s # && s > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 09 2020 *)

is(n)=n==1||vecmin(factor(n)[,1]%4)>1 \\ Charles R Greathouse IV, Apr 16 2012

list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=1,lim, if(u[i]==0, listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

A005089(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013

The number of terms below x is ~ (A * x / sqrt(log(x))) * (1 + C/log(x) + O(1/log(x)^2)), where A = A244659 and C = A244662 (Shanks, 1975). - Amiram Eldar, Jan 29 2022

More terms from Reinhard Zumkeller, Jun 17 2002

Name clarified by Evan M. Bailey, Sep 17 2019

A170818 a(n) is the product of primes (with multiplicity) of form 4*k+1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 1, 5, 1, 17, 1, 1, 5, 1, 1, 1, 1, 25, 13, 1, 1, 29, 5, 1, 1, 1, 17, 5, 1, 37, 1, 13, 5, 41, 1, 1, 1, 5, 1, 1, 1, 1, 25, 17, 13, 53, 1, 5, 1, 1, 29, 1, 5, 61, 1, 1, 1, 65, 1, 1, 17, 1, 5, 1, 1, 73, 37, 25, 1, 1, 13, 1, 5, 1, 41, 1, 1, 85, 1, 29, 1

Offset: 1

N. J. A. Sloane, Dec 22 2009

Completely multiplicative with a(p) = p if p = 4k+1 and a(p) = 1 otherwise. - Tom Edgar, Mar 05 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340.
Index to divisibility sequences

Cf. A170817-A170819, A097706, A083025, A072438, A286361.

a:= n-> mul(`if`(irem(i[1], 4)=1, i[1]^i[2], 1), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 1, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n)=my(f=factor(n)); prod(i=1,#f~,if(f[i,1]%4>1,1,f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import factorint, prod
def a072438(n):
    f = factorint(n)
    return 1 if n == 1 else prod(i**f[i] for i in f if i % 4 != 1)
def a(n): return n//a072438(n) # Indranil Ghosh, May 08 2017

a(n) = n/A072438(n). - Michel Marcus, Mar 05 2015

A286361 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+1} has: a(n) = A046523(A170818(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 6, 1, 1, 2, 1, 2, 1, 1, 2, 2, 4, 1, 1, 2, 1, 2, 1, 2, 1, 1, 6, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A170818, A267099, A267113, A286363, A286364, A286365.

Differs from A063014 for the first time at n=25, where a(25) = 4, while A063014(25) = 3.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a072438(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==1 else i**f[i] for i in f])
def a(n): return a046523(n/a072438(n)) # Indranil Ghosh, May 09 2017

(define (A286361 n) (A046523 (A170818 n)))

a(n) = A046523(A170818(n)).

a(n) = A286363(A267099(n)).

A072436 Remove prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 13, 2, 5, 16, 17, 2, 1, 20, 1, 2, 1, 8, 25, 26, 1, 4, 29, 10, 1, 32, 1, 34, 5, 4, 37, 2, 13, 40, 41, 2, 1, 4, 5, 2, 1, 16, 1, 50, 17, 52, 53, 2, 5, 8, 1, 58, 1, 20, 61, 2, 1, 64, 65, 2, 1, 68, 1, 10, 1, 8, 73, 74, 25, 4, 1, 26, 1, 80, 1, 82, 1, 4, 85, 2, 29

Offset: 1

Reinhard Zumkeller, Jun 17 2002

a(n) <= n; a(a(n)) = a(n); for all factors p^m of a(n): p=2 or p=4*k+1.

			a(90) = a(2*3*3*5) = a(2*(4*0+3)^2*(4*1+1)^1) = 2*1^2*5 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072438, A072437, A002144, A002145, A065338.

Equals n / A097706(n).

a:= n-> mul(`if`(irem(i[1], 4)=3, 1, i[1]^i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := n/Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, if ((f[k,1] % 4) == 3, f[k,1]=1)); factorback(f); \\ Michel Marcus, May 08 2017

from sympy import factorint
from operator import mul
def a(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])# Indranil Ghosh, May 08 2017

Multiplicative with a(p) = (if p==3 (mod 4) then 1 else p).

cp's OEIS Frontend

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

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A170818 a(n) is the product of primes (with multiplicity) of form 4*k+1 that divide n.

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A286361 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+1} has: a(n) = A046523(A170818(n)).

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A072436 Remove prime factors of form 4*k+3.

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